Abstract

We study the oscillation of all solutions of a general class of forced second-order differential equations,
where their second derivative is not necessarily a continuous function and the coefficients of the main equation may be
discontinuous. Our main results are not included in the previously published known oscillation criteria of interval type. Many
examples and consequences are presented illustrating the main results.

1. Introduction

Let and let denote the set of all real functions absolutely continuous on every bounded interval . We study the oscillatory behaviour of all solutions of the following class of forced second-order differential equations:
where the functions , , , and satisfy some general conditions given in Section 2. A continuous function is said to be oscillatory if there is a sequence , such that for all and as . A differential equation is oscillatory if all its solutions are oscillatory.

The forcing term is a sign-changing function (possibly discontinuous). This can be formulated by the following hypothesis: for every there exist two intervals and , , such that
The coefficient may be a discontinuous function on and the case occurs in our main results and examples too. Two important classes of functions are included in the differential operator as
The first one is the classic second-order differential operator which is linear in and the second one is the so-called one-dimensional mean curvature differential operator; see Examples 1 and 2.

Depending on , we propose the following four simple models for (1): (i) is strictly positive and continuous on as
(ii) is nonnegative and continuous on as
(iii) is nonnegative and discontinuous on as
(iv) is sign changing and discontinuous on as
where and is an arbitrary function such that for all , for instance, or . According to Corollaries 7 and 10, we will show that (4)–(7) are oscillatory provided the function satisfies for all ; see Examples 8–13. It is interesting that in particular for and , (4) allows an explicit oscillatory solution as shown in Figures 1 and 2.

Moreover, as a consequence of Corollary 7, one can show that all solutions of (4) are oscillatory; for details see Example 8. The main goal of this paper is to give some sufficient conditions on functions , and the coefficients , , and such that (1) is oscillatory; see Theorems 3 and 4. It will also cover the model equations (4)–(7) as well as some other examples presented in Section 2.

To the best of our knowledge, it seems that there are only few papers which study the oscillation of the second-order differential equations with nonsmooth (local integrable) coefficients; see [1–3]. More precisely, in [1] the author studied the interval oscillation criteria for the following second-order half-linear differential equation:
where and , such that . See also [2] but with the solution space instead of , that is, and .

In [3], the authors consider the following second-order differential equation:
where a.e. in , , and is locally integrable function in and continuous in . Equation (9) allows the forcing term in the next sense as follows:
where satisfies (2), but the functions and are smooth enough in their variables, that is, and .

On certain oscillation criteria for various classes of forced second-order differential equations with continuous coefficients, we refer the reader to [4–13]. Our method modifies a recently used one in [14, 15] and it contains the classic Riccati transformation of the main equation, a blow-up argument and pointwise comparison principle. The comparison principle applies to all sub- and supersolutions of a class of the generalized Riccati differential equations with nonlinear terms that are supposed to be locally integrable in the first variable and locally Lipschitz continuous in the second variable.

2. Hypotheses, Results, and Consequences

First of all, the function which appears in the second-order differential operator of (1) satisfies
where and is a locally Lipschitz function satisfying
In most cases, , , where , , and is an arbitrary function satisfying . Thus, for such with , condition (11) became:
It is not difficult to check that if or is a convex function, then it is locally Lipschitz on too; see for instance [16, Theorem ].

Two essential classes of the second-order differential operators satisfy condition (13), as is shown in the next examples.

Example 1. We consider the second-order differential operator which is linear in as follows:
where and for all . Obviously, the function satisfies condition (13) in particular for . Two usual choices for are and .

Example 2. We consider a quasilinear differential operator (the so-called one-dimensional prescribed mean curvature operator) as follows:
where and for all . It is not difficult to check that condition (13) is satisfied in particular for and for any . For , we can take the same choice as in the previous example.

Next, we suppose the existence of a constant such that
In order to simplify our consideration here, in many examples we often use .

Condition (2) means that there exists a sequence of pairs of intervals and , , contained in , such that the sequences , , , and are increasing, for each , and

On the intervals and , the coefficient satisfies

Let there be a real function , , and let there exist a sequence of positive real numbers such that
where , , and are constants defined in (11), (12), and (16), respectively.

The proof of the following main result will be presented in Section 4.

Theorem 3. Let the functions , , , and satisfy (11), (12), (16), (17), and (18), respectively. Let and on each interval , , . If (19) is fulfilled, then (1) is oscillatory.

Condition (19) can be replaced by an equivalent one, which has a more practical value and takes a simpler form since we do not need a sequence of auxiliary parameters : let there be a real function , such that
where , , and are constants defined in (11), (12), and (16), respectively. Since we will show that (19) and (20) are equivalent, see page 8, the next oscillation criterion immediately follows from Theorem 3.

Theorem 4. Let the functions , , , and satisfy (11), (12), (16), (17), and (18), respectively. Let and on each interval , , . If (20) is fulfilled, then (1) is oscillatory.

Remark 5. Assuming that , we can ensure the oscillation in the point . Note that since . Thus, we can generate a one-sided (right) limit.

Now, we consider some consequences of Theorem 4, which depend on the qualitative properties of the coefficient .

Substituting in (20), Theorem 4 implies the following result involving lower bounds on the lengths of intervals .

Corollary 6 ( is positive). Let the functions , , , and satisfy (11), (12), (16), (17), and (18), respectively. Let for each , . If
then (1) is oscillatory.

Corollary 7 ( is bounded from below by a positive constant). Let the functions , , , and satisfy (11), (12), (16), (17), and (18), respectively. Let there be two constants , satisfying
If
then (1) is oscillatory, where , , and are constants defined in (11), (12), and (16), respectively.

Example 8 8 (oscillation of (4)). We know that is an oscillatory solution of (4). However, according to Corollary 7, we can show that all solutions of (4) are oscillatory. Indeed, since , the conditions (11) and (12) are satisfied especially for , and . Next, implies that condition (16) is satisfied especially for . Since and , it is clear that conditions (18) and (22) are also satisfied in particular for and . Moreover, since and , , we have that (17) is fulfilled for , and . Moreover,
Hence, we conclude that the required condition (23) is fulfilled, that is,
Thus, all conditions of Corollary 7 are satisfied and hence (4) is oscillatory.

Example 9. We consider the following class of equations:
where , is an oscillatory function such that the zeros of the function satisfy , there is a such that for all , and on . This equation allows an explicitly given oscillatory solution . Moreover, if there is a constant such that
then by Corollary 7 we conclude that (26) is oscillatory. Indeed, conditions (11), (12), and (16) are satisfied by the same reasons as in Example 8. Condition (18) is satisfied because of . Also, from (27) it follows that (22) and (23) are fulfilled in particular for , , , , and , that is,
Hence Corollary 7 proves this result.

As the second consequence of Theorem 3 is unlike the first one, we consider the case when the coefficient is not a strictly positive function. Here by we denote the set of all such that .

Corollary 10 ( is nonnegative, but not ). Let the functions , , , and satisfy (11), (12), (16), (17), and (18), respectively. Let on each interval , , , such that
If
then (1) is oscillatory.

Proof. It suffices to show that (30) is equivalent to the existence of a real number such that
The claim will then follow from Theorem 3. Inequality (31) is for any equivalent to
that is, to
This inequality is easily seen to be equivalent to (30) for any . Note that if , then the second inequality in (31) is trivially satisfied.

Example 11 11 (oscillation of (5)). Equation (5) as well as equation
are oscillatory, where satisfies (16) with and , . Indeed, in (5) we have , and , and so, (17) and (18) are fulfilled, and , on each interval , , , where , , and
Moreover,
and since , , , and , for we have
Thus, the required condition (30) is fulfilled in this case and hence we may apply Corollary 10 to (5) to verify that this equation is oscillatory. Analogously, we can show that (34) is oscillatory too.

In the next example, the coefficient is a discontinuous function on .

Example 12 (oscillation of (6)). If satisfies (16) with , then (6) is oscillatory since the required condition (30) is satisfied especially for on , , and as in (35), , , , and . Indeed,
which shows that (30) is satisfied.

At the end of this section, we consider the case when changes sign on but with the help of Corollary 10 since is strictly positive on all intervals .

Example 13 (oscillation of (7)). If satisfies (16) with , then model equation (7) is oscillatory. In fact, let be as in (35). If , , it is clear that satisfies (2), that is, on and on . On the other hand, we have on and so , on for all , . Hence, the proof of the fact that all assumptions of Corollary 7 are fulfilled is the same as in the preceding example.

Let denote the set of all real functions which are absolutely continuous on every interval , where . For arbitrary numbers and functions and , we consider the ordinary differential equation
which generalizes the classic Riccati equation , where is an arbitrary function. We associate to (39) the corresponding sub- and supersolutions which satisfy, respectively,
We are interested in studying the following property:
In this way, we introduce the following definition.

Definition 14. We say that comparison principle holds for (39) on an interval , , if property (41) is fulfilled for all sub- and supersolutions of (39) on .

Now, we are able to state the next result.

Lemma 15. Let . Let and be two arbitrary function. For any , let there be a function depending on such that
Then comparison principle (41) holds for (39) on an interval , .

Proof. It is enough to use the idea of the proof of [14, Lemma 4.1]. Hence, this proof is left to the reader.

Corollary 16. Let , and let be a locally Lipschitz function on . Let be an arbitrary real number and an arbitrary function. Let . If on and , then comparison principle (41) holds for the Riccati differential equation

Proof. It is clear that (43) is a particular case of (39) in particular for
Since is a locally Lipschitz function on , for every there is an depending on such that
Hence, for any and all , we obtain:
Thus, according to assumption , the required condition (42) is fulfilled in particular for and so Lemma 15 proves this corollary.

Before we give the proof of Theorem 3, we state and prove the next two propositions. In the first one, by a nonoscillatory solution of the main equation (1), we get the existence of a supersolution of the Riccati differential equation (43) on the interval or . In the second one, we construct two subsolutions and of (43) which blow up on intervals and , respectively.

Proposition 17. Let and satisfy (11), (12), and (16), respectively, and let satisfy (2). Let , and on . Let be a nonoscillatory solution of (1) and let, for some and , the function be defined by
Then and satisfies the inequality
where if and if .

Proof. Since is a nonoscillatory solution of (1), there is a such that on . Hence, is well defined by (47).Because of (2), we have for all , where if and if and since we have
Next, since and are from , we can take the first derivative of for almost everywhere in , which together with , and on gives
that is,
Hence, (49) and (51) prove the desired inequality (48).

Proposition 18. Let (19) be satisfied. Let and be two arbitrary real numbers. Then there are two points and and two continuous functions and , such that

Proof. Since the function is a bijection from the interval into , we observe that there are two such that
Next, we define two functions and by
where the real numbers , and the function are defined in (19). From (19) and (54), one can immediately conclude that
Since , it follows that , which implies that , . Therefore, exploiting the fact that in (55) we have , we obtain the existence of two points and such that
Now, we are able to define the following two functions and by
That are well defined because of (56). Moreover, from (53), (54), and (56) we obtain
Also, since , we can take the first derivative of and hence from (19) and (57) we obtain
which together with (58) proves this proposition.

Now we are able to present the proof of Theorem 3 based on Lemma 15 and Propositions 17 and 18.

Proof of Theorem 3. Assuming the contrary, then there is a nonoscillatory solution and such that for all , and are from . In order to simplify notation, let every be fixed and omitted in the notation. For instance, instead of and we write and , respectively, and so on.On one hand, we observe by Proposition 17 that the function defined by (47) is a supersolution of the following Riccati differential equation:
where and if and if ; see the proof of Proposition 17. Let for instance and thus .On the other hand, by Proposition 18 there are a number and subsolutions , of the Riccati differential equation (60) such that
(in the case when , then we work with and the other subsolution , ). Hence by Corollary 16 and (61) we conclude that for all and therefore,
which contradicts . Thus, the assumption that is nonoscillatory is not possible and hence every solution of (1) is oscillatory.

Proof of Theorem 4. There is only one difference between Theorems 3 and 4, and it is the difference between conditions (19) and (20). Hence, Theorem 4 immediately follows from Theorem 3 provided that we prove the equivalence between (19) and (20).

First of all, we need the following two lemmas.

Lemma 19. Let , , and be positive real numbers, and . Then the existence of a positive real number such that
is equivalent to
Here is the conjugate exponent of .

Proof. Let us define . Since is decreasing and is increasing, there exists a unique point of maximum of . It is achieved at which is a solution of , hence .If there is a positive real number such that , then . Conversely, if , then , and the equivalence in the lemma is proved.

The preceding lemma is a special case of the following more general statement.

Lemma 20. Assume that , , and are positive real functions defined on a subset . Then the existence of a positive real number such that for all ,
is equivalent to

Proof. The condition that there exists such that (65) is true for all is equivalent with the following two inequalities, which have to hold simultaneously:
that is, with
Taking the supremum of the left-hand side and the infimum of the right-hand side, we obtain (66). Conversely, if (66) is satisfied, then, since the left-hand side in (66) cannot be equal to zero, while the right-hand side is less than , then inequality (66) defines a nonempty closed interval in (possibly reducing to just one point), in which we can choose any positive . Hence, (67) is satisfied for all , and therefore (65) as well.

Remark 21. If , , and , appearing in Lemma 20, are constant functions, then condition (66) is equivalent to , that is, to (64).

Proof of Equivalence of (19) and (20). We will use Lemma 20. Let us fix , where and . We see that that the inequality appearing in the second line of condition (19) is of the form (65), where
Condition (66) is equivalent to
This inequality is equivalent to the corresponding one in (19), and the claim follows from Lemma 19.

Proof of Corollary 6. Substituting instead of , and instead of in the inequality appearing in the second line of (20), after a short computation we obtain a stronger inequality than (20) as
Substituting we obtain that the left-hand side of (71) is equal to , and the resulting inequality is equivalent to (21). Since (21) implies (20), the claim is proved.

Remark 22. Here we show that the choice of in the proof of Corollary 6 is the best possible. To prove this, note that on the left-hand side of (71) we have the expression depending on an auxiliary function , and this function is absent on the right-hand side. Therefore, it has sense to try to find a function , , such that the value of is minimal (note that depends on the function as well). It is easy to see that the minimum is achieved for (or any positive constant). Indeed, since for any , by taking it suffices to assume that . The value of is minimal if is maximal possible, and since , it is clear that the maximum of is achieved when .

Proof of Corollary 7. It is clear that the condition (23) implies (21). Furthermore, since the lengths are uniformly bounded from below by a positive constant, see (23), then obviously as . Thus, the claim follows immediately from Corollary 6.

Proof of Corollary 10. Let . Then the required condition (19) is clearly satisfied because of assumption (30). Hence, this corollary immediately follows from Theorem 3.

In this section, we consider the oscillation of (1) in the case when is allowed in the assumption (12). In this way, the assumption (11) is slightly modified by a real number such that
where (12) is supposed, that is, for every and for a locally Lipschitz function for which there exists a constant such that for all .

For the function , we assume that there exists a constant such that

We will need the following two lemmas.

Lemma 23. Let , and let be a measurable function. Then,
if and only if there exists a function , , such that

Proof. We assume (74). If is finite, we may choose .For , we may define for every natural number a function
Since , we have . Obviously, is also infinite. This implies that there exists an such that . We define .Now, we assume (75). Integrating over the whole interval, we get

Lemma 24. Let . There exists a unique function that satisfies the following Cauchy problem:
Furthermore, there exists a finite real number such that
and is injective and monotonous. We have and is odd.

Proof. Obviously every solution to (78) is injective and monotonous on a connected set, since .Let . Obviously for all and for some constants . We may assume . Hence, , has (by [17, Theorem 3.1]) a locally unique solution on . By the same argument can be extended to the whole of , interval by interval of type for all .We know from [18] that the for the integral of the left-hand side we have
so .Since , is injective, monotonous, and of class and so we define . It immediately follows that and that is injective and monotonous. Also, by continuity of , we have .Differentiating, it also follows that and by definition of , . So, is a solution of (78)-(79). Since any other such solution must have as its inverse, is unique on its domain.

Proposition 25. Let and . Assume (72) and (73) and let and (or ) for all . If there exists a constant such that
then for any solution of (1) with (or ) there exists a number such that .